wp_delete_object_term_relationships() │ WP 2.3.0

Let's say there are tags "one" and "two" that are attached to post 567 and we need to remove this relationship (remove tags from the post):

<?php wp_delete_object_term_relationships( 567, 'post_tag' ); ?>

Delete multiple taxonomies’ relationships for a post

$post_id = 55;
$taxonomies = [ 'category', 'post_tag', 'custom_taxonomy' ];

wp_delete_object_term_relationships( $post_id, $taxonomies );

Suppose we have two taxonomies music and compositors and some posts that are in terms of both these taxonomies. And we need to select all the posts that are in the music taxonomy and remove them from the compositors taxonomy, if there is a relation to compositors at all.

Thus, we will split all the posts into two taxonomies and the post will be in either music or compositors.

del_tax_rel( $leave_in = 'music', $del_from = 'compositors' );

function del_tax_rel( $leave_in, $del_from ) {
	global $wpdb;

	$sql = "SELECT p.ID, post_title
	FROM $wpdb->posts p
		JOIN $wpdb->term_relationships rel ON (rel.object_id = p.ID)
		JOIN $wpdb->term_taxonomy tx ON (rel.term_taxonomy_id = tx.term_taxonomy_id)
	WHERE p.post_status = 'publish'
		AND tx.taxonomy = '$leave_in'";

	$results = $wpdb->get_results( $sql );

	if( ! $results ){
		return false;
	}

	foreach( $results as $pst ){
		wp_delete_object_term_relationships( $pst->ID, $del_from );
	}
}

Let's remove post 10 from term (category) 1, i.e. let's break the relation of the specified object (post, page) with an element of the taxonomy (term):

$object_id = 10;
$delete_terms = [ 1 ];

global $wpdb;

if( $delete_terms ){

	$in_delete_terms = "'" . implode( "', '", $delete_terms ) . "'";

	do_action( 'delete_term_relationships', $object_id, $delete_terms );

	$wpdb->query( $wpdb->prepare( 
		"DELETE FROM $wpdb->term_relationships WHERE object_id = %d AND term_taxonomy_id IN ($in_delete_terms)", 
		$object_id 
	) );

	do_action( 'deleted_term_relationships', $object_id, $delete_terms );

	wp_update_term_count( $delete_terms, $taxonomy );
}

You can specify several taxonomy elements in an array: [ 1, 2, 67 ].

This example doesn't exactly fit this function, but it is similar because it also breaks the relationship between an object and a taxonomy element. The difference is that here only 1 or more specified elements are broken, while wp_delete_object_term_relationships() breaks all relationships to the taxonomy at all.